Stability theory is a relatively new area of model theory which has developed rapidly. It is much concerned with types: with their properties and the relations between them, and with how this is reflected in terms of the structure of models. It is perhaps not surprising then, that one finds notions from stability theory relevant to modules (sometimes even identical with already existing algebraic notions). For we have seen that every type reduces essentially to a collection of pp formulas and negations of pp formulas, and pp formulas are not far from being “algebraic” since they express solvability of systems of linear equations.

I will state definitions and results from stability theory in various chapters, as I need them. A short introduction which covers most of what I will need is Pillay's book [Pi83]. A concise continuation of this is Makkai's article [Mak84]: Rather more inclusive are the books of Baldwin [Bal8?] and Lascar [Las8?]. The already-mentioned model theory texts of Poizat [Poi85] and Hodges [Ho??] contain all the stability theory that we will need. Of course there is also Shelah's book [She78], which contains a vast amount of material (though much is implicit rather than explicitly pointed out and the presentation is not designed to facilitate “dipping into” the book).

Stability theory divides complete theories into two major classes: those which are stable (where there is some possibility of developing a structure theory, at least for sufficiently saturated models); and those which are unstable (in the sense that they contain an infinite definable linear order and are, in some senses, less well-behaved than stable theories – though see [PiSt86]).